Returns to scale and the random production of innovations

I define constant, increasing and decreasing returns to scale in the production of innovations that occur randomly with a probability that depends upon resources spent in research. I analyse the mathematical representations of random processes of innovation that exhibit constant, increasing or decreasing returns to scale in that sense and determine their respective functional forms. I also give two complementary conditions, which are respectively sufficient for increasing returns to scale, and decreasing returns. Finally, as a particular case, I show processes that use only one factor of innovation and satisfy constant returns form a one-parameter family.